Projective limits of local shift morphism

TL;DR

This paper introduces the concept of projective limits of local shift morphisms, providing a differential structure and applying it to Poisson tensors, exemplified by the KdV equation on the circle.

Contribution

It defines projective limits of local shift morphisms and applies this framework to shift Poisson tensors, including a case study on the KdV equation.

Findings

Defined projective limits of local shift morphisms with differential structure

Characterized shift Poisson tensors as antisymmetric morphisms with vanishing Schouten bracket

Applied the framework to compatible Poisson tensors for the KdV equation on the circle

Abstract

We define the notion of projective limit of local shift morphisms of type $\left( r,s\right) $ and endow the space of such mathematical objects with an adapted differential structure. The notion of shift Poisson tensor $P$ on a Hilbert tower corresponds to such morphisms which are antisymmetric and whose Schouten bracket $\left[ P,P\right] $ vanishes. We illustrate this notion with the example of the famous KdV equation on the circle $\mathbb{S}^{1}$ for which one can associate a couple of compatible Poisson tensors of this type on the Hilbert tower $\left( H^{n}(\mathbb{S}^{1})\right) _{n\in\mathbb{N}% ^{\ast}}$.

Equations

$${\gamma }_i^j\circ f_j=f_i\circ {\delta }_i^j$$

$$\begin{array}[c]{cccc}f:&X&\rightarrow&Y\\ &\left(x_{i}\right)_{i\in\mathbb{N}}&\mapsto&\left(f_{i}\left(x_{i}\right)\right)_{i\in\mathbb{N}}\end{array}$$

$$R\left(t,v\right)=\left|\begin{array}[c]{cc}\dfrac{f\left(x+tv\right)-f\left(x\right)-Df_{x}\left(tv\right)}{t},&t\neq 0\\ 0,&t=0\end{array}\right.$$

$$\begin{array}[c]{cccc}Df:&U\times E&\longrightarrow&F\\ &\left(x,v\right)&\mapsto&Df_{x}\left(v\right)\end{array}$$

$$D{f}_x\left(h\right)=\underset{t⟶0}{\lim }\frac{f\left(x+th\right)-f\left(x\right)}{t}$$

$$\forall x=\left(x^i\right)\in U,D{f}_x=\underleftarrow{\lim }D{f}_{x^i}\text{.}$$

$$\forall n\in \mathbb{N},\forall \epsilon >0,\exists i_{\epsilon }\in \mathbb{N}:\forall \left(j,k\right)\in {\mathbb{N}}^2,k\ge j\ge i_{\epsilon },{\nu }_n\left(x_k-x_j\right)<\epsilon$$

$$\underset{i⟶+\infty }{\lim }{x}_i=x⟺\forall n\in \mathbb{N},\underset{i⟶+\infty }{\lim }{\nu }_n\left(x_i-x\right)=0$$

$${\nu }_n\left({\left(x_i\right)}_{i\in \mathbb{N}}\right)=\sum _{k=0}^n\left|x_k\right|$$

$$d\left(x,y\right)=\sum _{k=0}^{+\infty }\frac{\left|y_k-x_k\right|}{2^k\left(1+\left|y_k-x_k\right|\right)}$$

$$\forall x={\left(x_n\right)}_{n\in \mathbb{N}}\in \underleftarrow{\lim }{\mathbb{B}}_n,{\nu }_n\left(x\right)=\sum _{i=0}^n{\Vert x_n\Vert }_n$$

$$\begin{array}[c]{cccc}\pi_{i}^{j}:&\mathbb{F}_{j}&\longrightarrow&\mathbb{F}_{i}\\ &\left[x+\ker\nu_{j}\right]_{j}&\longmapsto&\left[x+\ker\nu_{i}\right]_{i}\end{array}$$

$$\forall n\in \mathbb{N},\exists k_n\in \mathbb{N},\exists C_n>0:\forall x\in {\mathbb{F}}_1,{\nu }_2^n\left(L.x\right)\le C_n{\nu }_1^{k_n}\left(x\right)$$

$$p_{n,B}\left(L\right)=\sup \left\{{\nu }_2^n\left(L.x\right),x\in B\right\}$$

$$\mathcal{H}\left({\mathbb{F}}_1,{\mathbb{F}}_2\right)=\left\{\left(L_n\right)\in \prod _{n\in \mathbb{N}}\mathcal{L}\left({\mathbb{B}}_1^n,{\mathbb{B}}_2^n\right):\underleftarrow{\lim }{L}_n\text{ exists}\right\}$$

$$\begin{array}[c]{ccc}\mathbb{B}_{1}^{i}&\overset{{\vphantom{A}}_{1}\delta_{i}^{j}}{\longleftarrow}&\mathbb{B}_{1}^{j}\\ L_{i}\downarrow&&\downarrow L_{j}\\ \mathbb{B}_{2}^{i}&\overset{{\vphantom{A}}_{2}\delta_{i}^{j}}{\longleftarrow}&\mathbb{B}_{2}^{j}\end{array}$$

$$\forall x\in {\mathbb{B}}_1^{n+r},{\Vert L.x\Vert }_2^{n-s}\le C_n{\Vert x\Vert }_1^{n+r}$$

$${\Vert L\Vert }_{L_n^{r,s}}=\underset{{\Vert x\Vert }_1^{n+r}}{\sup }{\Vert L.x\Vert }_2^{n-s}$$

$$\begin{array}[c]{cc}\mathcal{L}_{s,n}^{r,s}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right)=&\left\{\begin{array}[c]{c}\left(L_{s},\dots,L_{n}\right)\in\mathcal{L}_{s}^{r,s}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right)\times\cdots\times\mathcal{L}_{n}^{r,s}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right):\\ \forall\left(i,j\right)\in\mathbb{N}^{2}:n\geq j\geq i\geq s,{\vphantom{A}}_{2}\delta_{i-s}^{j-s}\circ L_{j}=L_{i}\circ{\vphantom{A}}_{1}\delta_{i+r}^{j+r}\end{array}\right\}\end{array}$$

$${\Vert \left(L_s,\dots ,L_n\right)\Vert }_{s,n}^{r,s}=\sum _{i=s}^n{\Vert L_i\Vert }_{L_i^{r,s}}$$

$$\begin{array}[c]{cccc}\pi_{i}^{j}:&\mathcal{L}_{s,j}^{r,s}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right)&\longrightarrow&\mathcal{L}_{s,i}^{r,s}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right)\\ &\left(L_{s},\dots,L_{j}\right)&\longmapsto&\left(L_{s},\dots,L_{i}\right)\end{array}$$

$$\begin{array}[c]{ll}\left\|\pi_{i}^{j}\left(L_{s},\dots,L_{j}\right)\right\|_{s,i}^{r,s}&=\left\|\left(L_{s},\dots,L_{i}\right)\right\|_{s,i}^{r,s}\\ &=\sum\limits_{k=s}^{i}\left\|L_{k}\right\|_{L_{k}^{r,s}}\\ &\leq\sum\limits_{k=s}^{j}\left\|L_{k}\right\|_{L_{k}^{r,s}}\\ &=\left\|\left(L_{s},\dots,L_{j}\right)\right\|_{s,j}^{r,s}\end{array}$$

$${\mathbb{F}}_n^0=\left\{\widehat{{\omega }_n}=\left({\omega }_0,\dots {\omega }_n\right)\in \prod _{i=0}^n{\mathbb{F}}_i^{{}^{\prime }}\right\}$$

$${\Vert \widehat{{\omega }_n}\Vert }^n=\sum _{i=0}^n\underset{{\Vert x_i\Vert }_i=1}{\max }\left|{\omega }_i\left(x_i\right)\right|$$

$$\phi :U_n⟶\mathcal{L}\left({\mathbb{F}}_2^{n+r},{\mathbb{F}}_3^{n-s}\right)$$

$$H^n\left({\mathbb{S}}^1\right)=\left\{q\in L^2\left({\mathbb{S}}^1\right):\forall k\in \left\{0,\dots ,n\right\},q^{\left(k\right)}\in L^2\left({\mathbb{S}}^1\right)\right\}$$

$${\phi }_n:U_n⟶\mathcal{L}\left(H_{n+r},H_{n-s}\right)$$

$$\begin{array}[c]{cccc}\partial_{x}:&U\cap H_{n}&\longrightarrow&\mathcal{L}\left(H_{n+1},H_{n}\right)\\ &q&\longmapsto&\left(\partial_{x}\right)_{q}\end{array}$$

$$\begin{array}[c]{cccc}\left(\partial_{x}\right)_{q}:&H_{n+1}&\longrightarrow&H_{n}\\ &u&\longmapsto&\partial_{x}u\end{array}.$$

$$\begin{array}[c]{cccc}L_{n}:&U\cap H_{n}&\longrightarrow&\mathcal{L}\left(H_{n+2},H_{n-1}\right)\\ &q&\longmapsto&\left(L_{n}\right)_{q}\end{array}$$

Full text

MSC2010: 46A13, 46E20, 46G05, 35R15.

Keywords: Local shift morphism; Projective limit; Shift Hilbert Poisson tensor; Inductive dual; Hilbert tower; KdV equation.

The author is grateful to Professor Fernand Pelletier for helpful comments.

1 Introduction

This article is dealing with projective limits of shift Poisson tensors on Hilbert towers whose set can be endowed with a Fréchet structure.

The paper is organized as follows. Section 2 introduces the basic notions and results on projective limits of Banach spaces and an adapted notion of differentiability on such spaces. In section 3, we introduce shift operators on direct limits of Banach spaces and endow the set of projective limits of such operators with a Fréchet structure (Theorem 3.4.3). In section 4, we introduce the notion of local shift morphism and study the smoothness of projective limits of such operators (Theorem 4.2.2). In section 5, we consider the particular case of Hilbert towers that appears as an adapted framework to describe some PDEs. Section 6 is devoted to the notion of shift Hilbert Poisson tensors $P$, corresponding to a projective limit of antisymmetric local shift morphisms defined on a Hilbert tower where the Schouten bracket $[P,P]$ vanishes. As a fundamental example, we consider the KdV equation on the circle $\mathbb{S}^{1}$ (cf. [KapMak]) for which there exists a couple of compatible shift Hilbert Poisson tensors on the projective limit of the Sobolev spaces $H^{n}\left(\mathbb{S}^{1}\right)$.

2 Projective limits of Banach spaces and differentiability

In a lot of situations in global analysis and Mathematical Physics, the framework of Banach or Hilbert spaces is not adapted any more. In some cases, the projective limits of such spaces must be adopted. For such Fréchet spaces, the differentiation method proposed by J.A. Leslie fits well to the requirements of this geometrical situation.

We can remark that the convenient setting, defined by A. Frölicher and A. Kriegl (see [FroKri] and [KriMic]), could have been used. This framework is adapted to various structures (e.g. for convenient partial Poisson structures as defined in [Pel]).

2.1 Projective limits of topological spaces

Definition 2.1.1**.**

$\left\{\left(X_{i},\delta_{i}^{j}\right)\right\}_{\left(i,j\right)\in\mathbb{N}^{2},\ j\geq i}$* is a projective sequence of topological spaces if*

(PSTS 1)

for all $i\in\mathbb{N},$ $X_{i}$ is a topological space;

(PSTS 2)

for all $i,$ $j$ $\in\mathbb{N},$ such that $j\geq i,$ $\delta_{i}^{j}:X_{j}\rightarrow X_{i}$ is a continuous mapping;

(PSTS 3)

for all $i\in\mathbb{N},$ $\delta_{i}^{i}={Id}_{X_{i}}$;

(PSTS 4)

for all integers $i\leq j\leq k$, $\delta_{i}^{j}\circ\delta_{j}^{k}=\delta_{i}^{k}$.

Definition 2.1.2**.**

*An element $\left(x_{i}\right)_{i\in\mathbb{N}}$ of the product $\prod\limits_{i\in N}X_{i}$ is called a thread if for all $j\geq i$, $\delta_{i}^{j}\left(x_{j}\right)=x_{i}$.

The set $X=\underleftarrow{\lim}X_{i}$ of such elements, endowed with the finest topology for which all the projections $\delta_{i}:X\rightarrow X_{i}$ are continuous, is called projective limit of the sequence $\left\{\left(X_{i},\delta_{i}^{j}\right)\right\}_{\left(i,j\right)\in\mathbb{N}^{2},\ j\geq i}$.*

A basis of the topology of $X$ is constituted by the subsets $\left(\delta_{i}\right)^{-1}\left(U_{i}\right)$ where $U_{i}$ is an open subset of $X_{i}$ (and so $\delta_{i}^{j}$ is open).

Definition 2.1.3**.**

*Let $\left\{\left(X_{i},\delta_{i}^{j}\right)\right\}_{\left(i,j\right)\in\mathbb{N}^{2},\ j\geq i}$ and $\left\{\left(Y_{i},\gamma_{i}^{j}\right)\right\}_{\left(i,j\right)\in\mathbb{N}^{2},\ j\geq i}$be two projective systems whose respective projective limits are $X$ and $Y$.

A sequence $\left(f_{i}\right)_{i\in\mathbb{N}}$ of continuous mappings $f_{i}:X_{i}\rightarrow Y_{i}$, satisfying for all $i,j\in\mathbb{N},$ $j\geq i,$ the condition*

[TABLE]

is called a projective system of mappings.

The projective limit of this sequence is the mapping

[TABLE]

The mapping $f$ is continuous and is a homeomorphism if all the $f_{i}$ are homeomorphisms (cf. [AbbMan]).

2.2 Differentiability

We first introduce the notion of differentiability à la Leslie between Hausdorff locally convex vector spaces $E$ and $F$ which corresponds to a particular case of the Gâteaux derivative. For full details, the reader is referred to [Les] and [DoGaVa]. Unlike the classical framework of Banach spaces, the derivative does not involve the space of continuous linear maps $\mathcal{L}\left(E,F\right)$ which has no reasonable structure.

Definition 2.2.1**.**

Let $E$ and $F$ be two Hausdorff locally convex vector spaces and let $U$ be an open subset of $E$. A continuous map $f:U\longrightarrow F$ is said to be differentiable at $x\in U$ if there exits a continuous linear map $Df_{x}:E\longrightarrow F$ such that

[TABLE]

is continuous at every $\left(0,v\right)\in\mathbb{R}\times F$. The map $Df_{x}$ is called the derivative (or differential) of $f$ at $x$.

The map is said to be differentiable if it is differentiable at every $x\in U$.

Note that, in this case, $Df_{x}$ is uniquely determined.

Definition 2.2.2**.**

A continuous map $f:U\longrightarrow F$ from an open subset $U$ of a Hausdorff locally convex vector space $E$ to a space of the same type $F$ is called C1-differentiable if it is differentiable at every $x\in U$, and if the derivative

[TABLE]

is continuous.

The notion of $C^{n}$-differentiability ($n\geq 2$) can be defined by induction (cf. [DoGaVa], Definition 2.2.3) and allows to define the $C^{\infty}$-differentiability à la Leslie which corresponds to the $C^{\infty}$-differentiability in the ordinary case.

We then have the following properties:

(PDL 1)

Every continuous linear map $f:E\longrightarrow F$ is Leslie $C^{\infty}$ and $Df=F$;

(PDL 2)

The differential at $x$ satisfies the relation

[TABLE]

(PDL 3)

The chain rules holds.

2.3 Differentiability on projective limits

The connection between projective limits of maps and differentiation is given by the following result ([DoGaVa], Propositions 2.3.11 and 2.3.12).

Proposition 2.3.1**.**

Let $\mathbb{F}_{1}=\underleftarrow{\lim}\mathbb{E}_{1}^{i}$ and $\mathbb{F}_{2}=\underleftarrow{\lim}\mathbb{E}_{2}^{i}$ projective limits of Banach spaces. Let also $f^{i}:U^{i}\longrightarrow\mathbb{E}_{2}^{i}$ be where, for all $i\in\mathbb{N}$, $U^{i}$ is an open set of $\mathbb{E}_{1}^{i}$. We assume that $U=\underleftarrow{\lim}U^{i}$ exists and is a non empty open subset of $\mathbb{F}_{1}$; we also assume that $f=\underleftarrow{\lim}f^{i}:U\longrightarrow\mathbb{F}_{2}$ exists. Then we have:

If each $f^{i}$ is differentiable (resp. smooth), then so is $f$ and

[TABLE]

3 Shift operators

In Analysis and Mathematical Physics, Banach representations break down. By weakening the topological requirement, replacing the norm by a sequence of semi-norms, one gets the notion of Fréchet space. For the subsections 3.1 (resp. 3.2), the reader is referred to [Bour], [RobRob] and [Tre] (resp. [DoGaVa]).

3.1 Fréchet spaces

Definition 3.1.1**.**

A Fréchet space is a Hausdorff, locally convex topological vector space that is metrizable and complete.

The topology of a Fréchet space $\mathbb{F}$ can be induced by a sequence of semi-norms $\left(\nu_{n}\right)_{n\in\mathbb{N}}$ that is complete with respect to such a sequence.

Recall that $\mathbb{F}$ is complete with respect to this topology if and only if every sequence $\left(x_{i}\right)_{i\in\mathbb{N}}$ in $\mathbb{F}$ is such that

[TABLE]

converges in $\mathbb{F}$ where the convergence in this Fréchet space is controlled by all the semi-norms $\nu_{n}$:

[TABLE]

Example 3.1.2**.**

The space of real sequences $\mathbb{R}^{\mathbb{N}}=\prod\limits_{n\in N}\mathbb{R}^{n}$ endowed with the usual topology is a Fréchet space where the corresponding sequence of semi-norms is given by

[TABLE]

Metrizability is defined from $d$ as follows

[TABLE]

and the completeness is inherited from that of each $\mathbb{R}$ of the infinite product.

The notion of Fréchet space is closely related with the projective limit of Banach spaces.

If $\left\{\left(\mathbb{B}_{n},\left\|\ \right\|_{n}\right)\right\}_{n\in\mathbb{N}}$ is a projective sequence of Banach spaces, then $\underleftarrow{\lim}\mathbb{B}_{n}$ is a Fréchet space (cf. [DoGaVa], Theorem 2.3.7) where the sequence $\left(\nu_{n}\right)_{n\in\mathbb{N}}$ of semi-norms is given by

[TABLE]

Conversely, if $\mathbb{F}$ is a Fréchet space with associated semi-norms $\nu_{n}$, the completion $\mathbb{F}_{n}$ of the normed space $\mathbb{F}/\ker\nu_{n}$ is a Banach space called the local Banach space associated to the semi-norm $\nu_{n}$. It will be denoted by $\left(\mathbb{F}_{n},\left\|\ \right\|_{n}\right)$ where $\left\|\ \right\|_{n}$ is the norm associated to $\nu_{n}$. We then get a projective system $\left\{\left(\mathbb{F}_{i},\pi_{i}^{j}\right)\right\}_{\left(i,j\right)\in\mathbb{N}^{2},\ j\geq i}$ of Banach spaces whose bonding maps are

[TABLE]

where the bracket $\left[\ \ \right]_{n}$ corresponds to the associated equivalence class. $\mathbb{F}$ will be identified with the projective limit $\underleftarrow{\lim}\mathbb{F}_{i}$ (cf. [DoGaVa], Theorem 2.3.8).

The representation of Fréchet spaces as projective limits of Banach spaces is very interesting: Issues arising in the Fréchet framework can be solved by considering their components in the Banach factors of the associated projective sequence. So different pathological entities in the Fréchet framework can be replaced by approximations compatible with the inverse limits, e.g. ILB-Lie groups ([Omo]) or projective limits of Banach Lie groups ([Gal1]), manifolds ([AbbMan]), bundles ([Gal2], [AghSur]), algebroids ([Cab]), connections and differential equations ([ADGS]).

3.2 The Fréchet space $\mathcal{H}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right)$

Let $\mathbb{F}_{1}$ (resp. $\mathbb{F}_{2}$)  be a Fréchet space and $\left(\nu_{1}^{n}\right)_{n\in\mathbb{N}}$ (resp. $\left(\nu_{2}^{n}\right)_{n\in\mathbb{N}}$) the sequence of semi-norms of $\mathbb{F}_{1}$(resp. $\mathbb{F}_{2}$).

Recall ([Vog], 2.) that a linear map $L:\mathbb{F}_{1}\longrightarrow\mathbb{F}_{2}$ is continuous if

[TABLE]

The space $\mathcal{L}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right)$ of continuous linear maps between both these Fréchet spaces generally drops out of the Fréchet category. Indeed, $\mathcal{L}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right)$ is a Hausdorff locally convex topological vector space whose topology is defined by the family of semi-norms $\left\{p_{n,B}\right\}$:

[TABLE]

where $n\in\mathbb{N}$ and $B$ is any bounded subset of $\mathbb{F}_{1}$ containing $0_{\mathbb{F}_{1}.}$ This topology is not metrizable since the family $\left\{p_{n,B}\right\}$ is not countable.

So $\mathcal{L}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right)$ will be replaced, under certain assumptions, by a projective limit of appropriate functional spaces as introduced in [Gal2].

If we denote by $\mathcal{L}\left(\mathbb{B}_{1}^{n},\mathbb{B}_{2}^{n}\right)$ the space of linear continuous maps (or equivalently bounded linear maps because $\mathbb{B}_{1}^{n}$ and $\mathbb{B}_{2}^{n}$ are normed spaces), we then have the following result ([DoGaVa], Theorem 2.3.10).

Theorem 3.2.1**.**

The space of all continuous linear maps between $\mathbb{F}_{1}$ and $\mathbb{F}_{2}$ that can be represented as projective limits

[TABLE]

is a Fréchet space.

For this sequence $\left(L_{n}\right)$ of linear maps, for any integer $0\leq i\leq j$, the following diagram is commutative

[TABLE]

3.3 Shift operators

We assume that $\mathbb{F}_{1}=\underleftarrow{\lim}\mathbb{B}_{1}^{n}$ (resp. $\mathbb{F}_{2}=\underleftarrow{\lim}\mathbb{B}_{2}^{n}$) is a Fréchet space where $\left\{\left(\mathbb{B}_{1}^{i},_{{\vphantom{A}}1}\delta_{i}^{j}\right),\left\|\ \right\|_{1}^{i}\right\}_{\left(i,j\right)\in\mathbb{N}^{2},\ j\geq i}$(resp. $\left\{\left(\mathbb{B}_{2}^{i},_{{\vphantom{A}}2}\delta_{i}^{j}\right),\left\|\ \right\|_{2}^{i}\right\}_{\left(i,j\right)\in\mathbb{N}^{2},\ j\geq i}$) is a projective sequence of Banach spaces.

Definition 3.3.1**.**

A linear map $L:\mathbb{B}_{1}^{n+r}\longrightarrow\mathbb{B}_{2}^{n-s}$ is called a shift operator of base $n$ and type $\left(r,s\right)\in\mathbb{N}\times\mathbb{N}$ where $n\geq s$, if there exists $C_{n}>0$ such that:

[TABLE]

Notation 3.3.2**.**

$\mathcal{L}_{n}^{r,s}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right)$* denotes the set of shift operators of base $n$ and type $\left(r,s\right)$.*

Lemma 3.3.3**.**

$\mathcal{L}_{n}^{r,s}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right)$* endowed with the norm $\left\|\ \right\|_{L_{n}^{r,s}}$ defined by*

[TABLE]

is a Banach space.

A linear operator of base $n$ and type $\left(r,s\right)$ is continuous.

Example 3.3.4**.**

([Ham], 1.1.2, Examples (4) and 1.2.3 Examples (3)). Let $X$ be a compact manifold. Then $C^{\infty}\left(X\right)$ is a Fréchet space and for any linear partial differential operator $L$ of degree $r,$ we have $\left\|L.f\right\|_{n}\leq\left\|f\right\|_{n+r}$; so $L$ is a shift operator of base $n$ and type $\left(r,0\right)$ (tame operator in Hamilton’s terminology).

3.4 Projective limit of shift operators

Lemma 3.4.1**.**

For any integer $n\geq s$, the following set

[TABLE]

can be endowed with a structure of Banach space relatively to the norm $\left\|\ \right\|_{s,n}^{r,s}$ defined by

[TABLE]

Proof.

Since $\mathcal{L}_{s,n}^{r,s}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right)$ is a closed subspace of the Banach space $\mathcal{L}_{s}^{r,s}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right)\times\cdots\times\mathcal{L}_{n}^{r,s}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right)$, it is also a Banach space. ∎

Lemma 3.4.2**.**

For $j\geq i\geq s$, the canonical projections

[TABLE]

are linear and continuous.

Proof.

For $j\geq i\geq s$, the linearity of $\pi_{i}^{j}$ is obvious.

The continuity of $\pi_{i}^{j}$ is a consequence of

[TABLE]

∎

We then have the following result.

Theorem 3.4.3**.**

$\left\{\left(\mathcal{L}_{s,i}^{r,s}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right),\pi_{i}^{j}\right)\right\}_{\left(i,j\right)\in\mathbb{N}^{2},\ j\geq i\geq s}$is a projective sequence of Banach spaces whose projective limit $\mathcal{L}^{r,s}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right)$ can be endowed with a Fréchet structure.

Proof.

For $k\geq j\geq i\geq s$, it is obvious that $\pi_{i}^{k}=\pi_{i}^{j}\circ\pi_{j}^{k}$. Thus, according to Lemma 3.4.1 and Lemma 3.4.2, $\left\{\left(\mathcal{L}_{s,i}^{r,s}\left(\mathbb{F}_{1},\mathbb{F}_{2}\right),\pi_{i}^{j}\right)\right\}_{\left(i,j\right)\in\mathbb{N}^{2},\ j\geq i\geq s}$ is a projective sequence of Banach spaces. So its projective limit can be endowed with a structure of Fréchet space (cf. 3.1). ∎

3.5 Inductive dual

Because the dual of a Fréchet space generally drops out of the Fréchet category, it will be replaced by the inductive dual which is defined as a projective limit of Banach spaces.

Let $\mathbb{F}$ be a graded Fréchet space and let $\left(\mathbb{F}_{n}\right)_{n\in\mathbb{N}}$ be the sequence of associated Banach spaces. We then consider, for $n\in\mathbb{N}$, the following space

[TABLE]

where $\mathbb{F}_{i}^{{}^{\prime}}$ is the topological dual of the Banach space $\mathbb{F}_{i}$. Then $\mathbb{F}_{n}^{0}$ is a Banach space for the norm $\left\|\ \right\|^{n}$ defined by

[TABLE]

Definition 3.5.1**.**

The projectif limit of the sequence $\left\{\left(\mathbb{F}_{n}^{0},\Pi_{n}^{n+1}\right)\right\}_{n\in\mathbb{N}^{\ast}}$, where $\Pi_{n}^{n+1}:\mathbb{F}_{n+1}^{0}\longrightarrow\mathbb{F}_{n}^{0}$ is the natural projection, is called the inductive dual of $\mathbb{F}$ et denoted by $\mathbb{F}^{0}$.

The inductive dual $\mathbb{F}^{0}$ is a graded Fréchet space.

The inductive cotangent bundle $T^{0}\mathbb{F}$ is defined as the trivial bundle of base $\mathbb{F}$ and fiber $\mathbb{F}^{0}$ and appears as as the projective limit of $\left(\mathbb{F}_{n}\times\mathbb{F}_{n}^{0},\left\|\ \right\|_{n}+\left\|\ \right\|^{n}\right)$. An inductive differential form is a smooth section of this bundle.

4 Projective sequence of local shift morphisms

4.1 Local shift morphisms

Let $\mathbb{F}_{1}$ (resp. $\mathbb{F}_{2},\mathbb{F}_{3}$) be a graded Fréchet space and let $\left(\mathbb{F}_{1}^{n},\left\|\ \right\|_{n}^{1}\right)_{n\in\mathbb{N}}$ (resp. $\left(\mathbb{F}_{2}^{n},\left\|\ \right\|_{n}^{2}\right)_{n\in\mathbb{N}},\left(\mathbb{F}_{3}^{n},\left\|\ \right\|_{n}^{3}\right)_{n\in\mathbb{N}}$) be the sequence of associated local Banach spaces.

Definition 4.1.1**.**

Let $n\in\mathbb{N}$ such that $n-s\geq 0$. A smooth map

[TABLE]

where $U_{n}$ is an open set of $\mathbb{F}_{1}^{n}$, is called a local shift morphism of base $n$ and type $\left(r,s\right)\in\mathbb{N}\times\mathbb{N}$ above $U_{n}$.

4.2 Projective sequence of local shift morphisms

Definition 4.2.1**.**

A sequence $\left(\varphi_{n}\right)_{n\in\mathbb{N},\ n\geq s}$ of local shift morphisms $\varphi_{n}$ of type $\left(r,s\right)\in\mathbb{N}\times\mathbb{N}$ above $U_{n}$ is said to be a projective sequence of local shift morphisms if

(PSLSM 1)

$U_{s}\supset U_{s+1}\supset\cdots\supset U_{n}\supset U_{n+1}\supset\cdots$* and $U=\bigcap\limits_{n=s}^{+\infty}U_{n}$ is a non empty open set of $\mathbb{F}_{1}$;*

(PSLSM 2)

For any $q=(q_{n})_{n\in\mathbb{N}}\in U$, we have the following commutative diagram:

\textstyle{U_{n}\times\mathbb{F}_{2}^{n+r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\operatorname{Id}_{U_{n}},\,\varphi_{n}(q_{n}))\;\;\;}$$\textstyle{U_{n+1}\times\mathbb{F}_{2}^{n+r+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\vphantom{A}}_{1}\delta_{n}^{n+1}\times{\vphantom{A}}_{2}\delta_{n+r}^{n+r+1}}$$\scriptstyle{(\operatorname{Id}_{U_{n+1}},\,\varphi_{n+1}(q_{n+1}))\;\;\;}$$\scriptstyle{{\vphantom{A}}_{2}\pi_{n+1}^{n+1+r}}$$\textstyle{U_{n}\times\mathbb{F}_{3}^{n-s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\vphantom{A}}_{3}\pi_{n}^{n-s}}$$\textstyle{U_{n+1}\times\mathbb{F}_{3}^{n-s+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\;\;{\vphantom{A}}_{1}\delta_{n}^{n+1}\times{\vphantom{A}}_{3}\delta_{n-s}^{n-s+1}}$$\textstyle{U_{n}}$$\textstyle{U_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\vphantom{A}}_{1}\delta_{n}^{n+1}} **

Theorem 4.2.2**.**

The projective limit $\underleftarrow{\lim}\varphi_{n}$ of a projective sequence of local shift morphisms $\varphi_{n}$ of type $\left(r,s\right)\in\mathbb{N}\times\mathbb{N}$ above $U_{n}$ is a smooth map from the open set $U=\bigcap\limits_{n=s}^{+\infty}U_{n}$ of the Fréchet space $\mathbb{F}_{1}$ to the Fréchet space $\mathcal{L}^{r,s}\left(\mathbb{F}_{2},\mathbb{F}_{3}\right)$.

Proof.

Since $\mathcal{L}^{r,s}\left(\mathbb{F}_{2},\mathbb{F}_{3}\right)$ is the projective limit of the Banach spaces $\mathcal{L}_{n}^{r,s}\left(\mathbb{F}_{2},\mathbb{F}_{3}\right)$ (cf. Theorem 3.4.3) the smoothness of $\underleftarrow{\lim}\varphi_{n}$ results from the smoothness of the maps $\varphi_{n}$ and the Proposition 2.3.1. ∎

5 Hilbert towers

In this section, the reader is referred to [KapMak].

We consider the particular case where the Fréchet spaces $\mathbb{F}_{1}$, $\mathbb{F}_{2}$ and $\mathbb{F}_{3}$ are all equal to a same projective limit of Hilbert spaces.

5.1 Definition. Example

Definition 5.1.1**.**

The sequence $\left(H_{n}\right)_{n\in\mathbb{N}}$ is a Hilbert tower if

(HT 1)

$\left(H_{n}\right)_{n\in\mathbb{N}}$* is a decreasing sequence of Hilbert spaces: $H_{0}\supset H_{1}\supset\cdots$;*

(HT 2)

$\forall n\in\mathbb{N},\ \overline{H_{n+1}}=H_{n}$;

(HT 3)

There exists a basis of $H_{\infty}=\bigcap\limits_{n\in\mathbb{N}}H_{n}$, i.e. an orthonormal basis $\left(e_{m}\right)_{m\in\mathbb{N}}$ of $H_{0}$, where $e_{m}\in H_{\infty}$, such that $\left(e_{m}\right)_{m\in\mathbb{N}}$ is a basis of any $H_{N}$ (with $N\in\mathbb{N}$).

A Hilbert tower can be seen as an IHL space as defined in [Omo].

Example 5.1.2**.**

The sequence of Sobolev spaces $\left(H^{n}\left(\mathbb{S}^{1}\right)\right)_{n\in\mathbb{N}}$ where

[TABLE]

is a Hilbert tower where the orthonormal basis is $\left(e_{0},e_{1},e_{-1},\dots,e_{k},e_{-k},\dots\right)$, ($k\in\mathbb{N}$) where $e_{k}:x\mapsto e^{i2k\pi x}$.

Let $\left(H_{n}\right)_{n\in\mathbb{N}}$ be a Hilbert tower where $\iota_{n}^{n+1}:H_{n+1}\longrightarrow H_{n}$ is the natural injection and let us denote $\left\langle.,.\right\rangle_{n}$ the inner product of $H_{n}$ and $\left\|\ \right\|_{H_{n}}$ the associated norm.

The projective limit $H_{\infty}$ of the Hilbert tower $\left(H_{n}\right)_{n\in\mathbb{N}}$ is perfectly defined and can be endowed with a structure of Fréchet space.

5.2 Local shift Hilbert morphisms

In the sequel, we reformulate some of the precedent results in the particular case of a Hilbert tower $\left(H_{n}\right)_{n\in\mathbb{N}}$, that is for all $n\in\mathbb{N},\mathbb{F}_{1}^{n}=\mathbb{F}_{2}^{n}=\mathbb{F}_{2}^{n}=H_{n}$, where the norm $\left\|\ \right\|_{1}^{n}=\left\|\ \right\|_{2}^{n}=\left\|\ \right\|_{3}^{n}=\sqrt{\left\langle.,.\right\rangle_{n}}$ are associated to the inner product of $H_{n}$.

Definition 5.2.1**.**

A local shift Hilbert morphism of base $n$ and type $\left(r,s\right)$ is a smooth map

[TABLE]

where $U_{n}$ is an open set of $H_{n}$.

Example 5.2.2**.**

On the Sobolev tower $\left(H_{n}=H^{n}\left(\mathbb{S}^{1}\right)\right)_{n\in\mathbb{N}}$ (cf. Example 5.1.2), we consider the operator

[TABLE]

which corresponds to the first Poisson structure for the KdV equation (cf. section LABEL:_ExampleKdVEquationS1.) where $U=H_{0}=H^{0}\left(\mathbb{S}^{1}\right)$ and

[TABLE]

So $\partial_{x}$ is a local shift Hilbert morphism of type $\left(1,0\right)$ above any $H_{n}=H^{n}\left(\mathbb{S}^{1}\right)$.

Example 5.2.3**.**

On the Sobolev tower $\left(H^{n}\left(\mathbb{S}^{1}\right)\right)_{n\in\mathbb{N}}$, the operator

[TABLE]

corresponds to the second Poisson structure for the KdV equation where $U=H_{0}=H^{0}\left(\mathbb{S}^{1}\right)$ and

[TABLE]

$L_{n}$* is then a local shift morphism of type $\left(2,1\right)$ above $H_{n}=H^{n}\left(\mathbb{S}^{1}\right)$.*

In particular, we have, for $q\in H^{n}\left(\mathbb{S}^{1}\right)$,

[TABLE]

because

[TABLE]

where the norm $\left\|\ \right\|_{n}$ is given by

[TABLE]

5.3 Projective limits of local shift Hilbert morphisms

Definition 5.3.1**.**

Let $\left(H_{n}\right)_{n\in\mathbb{N}}$ be a Hilbert tower. A sequence $\left(\varphi_{n}\right)_{n\in\mathbb{N},\ n\geq s}$ of local shift morphisms $\varphi_{n}$ of type $\left(r,s\right)\in\mathbb{N}\times\mathbb{N}$ above $H_{n}$ is said to be a projective sequence of local shift Hilbert morphisms if, for any $q=(q_{n})\in\prod\limits_{n\in N}H_{n}$, we have the following commutative diagram:

[TABLE]

Let $\left(H_{n}\right)_{n\in\mathbb{N}}$ be a Hilbert tower and consider $H_{\infty}=\bigcap\limits_{n\in\mathbb{N}}H_{n}=\underleftarrow{\lim}H_{n}$. For $n\geq s$, the space

[TABLE]

is a Banach space. We then get a projective sequence $\left\{\left(\mathcal{H}_{s,i}^{r,s}\left(H_{\infty}\right),\pi_{i}^{j}\right)\right\}_{\left(i,j\right)\in\mathbb{N}^{2},\ j\geq i\geq s}$ where

[TABLE]

Its projective limit $\mathcal{H}^{r,s}\left(H_{\infty}\right)$ can be endowed with a structure of Fréchet space

For a projective sequence of local shift Hilbert morphisms $\left(\varphi_{n}\right)_{n\in\mathbb{N},\ n\geq s}$ of type $\left(r,s\right)$, we have the following commutative diagram:

[TABLE]

where the maps $\left(\varphi_{s},\dots,\varphi_{n}\right):U\cap H_{s}\times\cdots\times U\cap H_{n}\longrightarrow\mathcal{H}_{s,n}^{r,-s}\left(H_{\infty}\right)$ are smooth.

We can define the projective limit

[TABLE]

and this limit is smooth.

Example 5.3.2**.**

The sequence $\left(L_{n}\right)_{n\in\mathbb{N}}$ of Example 5.2.3 is a projective sequence of local shift morphisms of type $\left(2,1\right)$.

6 Shift Hilbert Poisson tensors

The notion of Poisson tensor is relevant in Mechanics and Mathematical Physics. It corresponds to a tensor field $P$ twice contravariant whose Schouten bracket $\left[P,P\right]$ vanishes. Bihamiltonian structures corresponding to a pair of compatible Poisson tensors is a fundamental tool in the resolution of some dynamical systems because the recursion operator linking both structures gives rise to conservation laws.

In the framework of Hilbert towers, thanks to the identification of a Hilbert space with its dual (Riesz Theorem), the morphism $P$ from the cotangent bundle to the tangent bundle can be seen as a projective limit of local shift Hilbert morphisms. Such objects are adapted to the description of the KdV equation on the circle $\mathbb{S}^{1}$.

Definition 6.0.1**.**

*Let $\left(P_{n}\right)_{n\in\mathbb{N}}$ be a sequence of local shift morphisms of type $\left(r,s\right)$ on the Hilbert tower $\left(H_{n}\right)_{n\in\mathbb{N}}$ whose projective limit is $P=\underleftarrow{\lim}P_{n}$.

$P$ is said to be a shift Hilbert Poisson tensor of type $\left(r,s\right)$ on $H_{\infty}=\underleftarrow{\lim}H_{n}$ if, for any $q=\underleftarrow{\lim}q_{n}$, $f=\underleftarrow{\lim}f_{n}$, $g=\underleftarrow{\lim}g_{n}$ and $h=\underleftarrow{\lim}h_{n}$, it fulfils the following conditions:*

(SHPT 1)

$P$* is antisymmetric,*

i.e. for all $n\in\mathbb{N}$ such that $n-s\geq 0$,

[TABLE]

(SHPT 2)

The Schouten bracket vanishes: $\left[P,P\right]=0$,

where for all $n\in\mathbb{N}$ such that $n+r-2s\geq 0$,

[TABLE]

In this definition, the differentiabity of $P$ at $q$ is given by:

[TABLE]

Example 6.0.2**.**

*The Korteweg-de Vries (KdV) equation ([KorVri]) is an evolution equation in one space dimension which was proposed as a model to describe waves on shallow water surfaces. This nonlinear and dispersive PDE was first introduced by J. Boussinesq ([Bous]) and rediscovered by D. Korteweg and G. de Vries ([KorVri]) in order to modelize natural phenomena discovered by Russel ([Rus]).

In [Arn], V. Arnold suggested a general framework for the Euler equations on an arbitrary group that describe a geodesic flow with respect to a suitable one-sided invariant Riemannian metric on the group. This approach works for the Virasoro group and provides a natural geometric setting for the KdV equation (cf. [KheMis]).*

It is well known (e.g. [FMPZ], [MagMor], [Olv], [Sch], [ZubMag], …) that this equation can be written in Hamiltonian form in two distinct ways. Moreover, there exists an infinite hierarchy of commuting conservation laws and Hamiltonian flows generated by a recursion operator linking both Poisson brackets. Such an equation can be viewed as a complete integrable system and has a lot of remarkable properties, including soliton solutions. In [KisLeu], the framework of variational Lie algebroids is used to describe such an evolutionary equation.

Here we consider the KdV equation on the circle $\mathbb{S}^{1}$ of unit length

[TABLE]

*where $t\in\mathbb{R}$ and $x\in\mathbb{S}^{1}.$

This equation can be seen as an infinite dimensional system on the Hilbert tower $\left(H^{n}\left(\mathbb{S}^{1}\right)\right)_{n\in\mathbb{N}}$ (cf. [KapMak] and [KapPos]). This system can be written in a bihamiltonian way relatively to the compatible shift Hilbert Poisson tensors $\partial_{x}$, of type $\left(1,0\right)$, and $L_{q}$ of type $\left(2,1\right)$.